Squaring of numbers near the base
- 1. SQUARING OF NUMBERS NEAR THE BASE. Observe simple examples: 97 Squared = 97 - 3 / 3x3 = 94 / 09 96 Squared = 96 - 4 / 4x4 = 92 / 16. When the number being squared is above the base, of 100 here, we add the Excess and Square the Excess: 104 Squared = 104 + 4 / 4x4 = 108 / 16 = 10, 816 104 x 105 = 104 + 5 / 4x5 = 109 / 20 = 10, 920 What if we enlarged our numbers to 998 Squared? It is close to 1,000 so we say Base 1,000 and know to have 3 zero's on the right hand side of the ( / ). 998 Squared = 998 - 2 / 2x2 = 996 / _ _ 4 = 996 / 004. = 996,004 Understanding this, you can be calculating digits in the millions: 9998 Squared = 9998 - 2 / 2x2 = 9996 / _ _ _ 4 (Since we are in Base 10,000 the 4 zeros determine the need for 4 digits after the ( / ). = 9996 / 0004 = 99,960,004. THE SQUARING OF NUMBERS ENDING IN 5 If we wanted to square the number 25, ie 25 x 25, we would conventionally take 3 lines of working out. Vedic Mathematics merely looks at the Question, applys one of the 16 Sutras, and solves it Mentally in One-Line. In this case, the Sutra at work is "By One More Than The Previous Digit". Observing that 25 is a 2 digit number, 5 is the last digit, but we are mainly interested in "the Previous Digit" which is 2. We say, mentally, "What is One more than Two?" It is 3. The word "By" in the Sutra really means "to multiply". The setting out for the first half of the answer is thus: 25 Squared = 2 "By" 3 / ........ = 2 x 3 / ........ To this we tag on the last digit "5" squared: = 2 x 3 / 5 x 5 = 6 / 25 Thus the answer is 625.
- 2. Similarly, all other numbers that end in 5, when squared, can be done instantly: 15 Squared = 1 x 2 / 5 x 5 = 2 / 25 = 225 35 Squared = 3 x 4 / 5 x 5 = 12 / 25 = 1,225 45 Squared = 4 x 5 / 5 x 5 = 20 / 25 = 2,025 95 Squared = 9 x 10 / 5 x 5 = 90 / 25 = 9, 025